On the three-dimensional light scattering by a large nonspherical particle based on vectorial complex ray model

(1)

HAL Id: tel-02975280

https://tel.archives-ouvertes.fr/tel-02975280

Submitted on 22 Oct 2020

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

nonspherical particle based on vectorial complex ray

model

Qingwei Duan

To cite this version:

Qingwei Duan. On the three-dimensional light scattering by a large nonspherical particle based on vectorial complex ray model. Fluid mechanics [physics.class-ph]. Normandie Université; Xidian University (Xi’an (Chine)), 2020. English. �NNT : 2020NORMR018�. �tel-02975280�

(2)

(3)

(4)

(5)

(6)

分分分类类类号号号 O435 密密密级级级公公公开开开

西

西安

安

安电

电

电子

子科

子

科

科技

技

技大

大

大学

学

博

博士

士

士学

学

学位

位

位论

论

论文

文

基

基于

于

于矢

矢

矢量

量

量复

复射

复

射

射线

线

线模

模

模型

型

型的

的

大

大尺

尺

尺寸

寸

寸非

非球

非

球

球形

形

形粒

粒

粒子

子

子三

三维

三

维

维光

光

光散

散

散射

射

射场

场

场计

计

计算

算

算方

方

方法

法

作

作者

者

者姓

姓

姓名

名

名：

：

：段庆威

一

一级

级

级学

学

学科

科

科：

：

：物理学

二

二级

级

级学

学

学科

科

科（

（

（研

研

研究

究

究方

方

方向

向

向）

）

：

：光学

学

学位

位

位类

类

类别

别

别：

：

：理学博士

指

指导

导

导教

教

教师

师

师姓

姓

姓名

名

名、

、

、职

职

职称

称

称：

：

：韩香娥，任宽芳教授

学

院

院：

：

：物理与光电工程学院

提

提交

交

交日

日

日期

期

期：

：

：2020年6月

法国鲁昂大学联合培养

(7)

(8)

Acknowledgements

This thesis would not have been completed without much guidance, encouragement and support from many people. I want to take advantage of this opportunity to express my cordial thanks.

Firstly, I would like to extend my greatest gratitude to professors Xiang’e Han (China) and Kuan Fang Ren (France), my advisors and supervisors, for their insightful guidance and patient cultivation. Their rigorous attitude towards academic research, strong sense of responsibility and enthusiasm for work have affected me deeply during the past years. This thesis contains their invaluable instructions and support, from beginning to end.

This thesis was completed during my studies in the Xidian University (China) and the Coria institute, Universit´e de Rouen Normandie (France). I would like to express my sincere thanks to Prof. Lixin Guo, Prof. Bing Wei, Prof. Lu Bai, Prof. Yiping Han, Prof. Ruike Yang, Prof. Min Zhang, Prof. Hongfu Guo, associate Profs. Renxian Li and Jiajie Wang for their guidance and help during my study in Xidian University.

I want to acknowledge the financial support from the China Scholarship Council (CSC) for my study in France as a joint PhD. student.

I appreciate the fruitful discussions with professor Fabrice R.A. Onofri of the Aix-Marseille Université on my experimental and theoretical work. In addition, I would like to acknowledge the patience from Claude Rozé and Lo¨ıc Méès for being the Monitoring Committee members (CSI) of this thesis and for assessing the progress and quality of the work. Many thanks to Dr. Ruiping Yang and Dr. Sa¨ıd Idlahcen of the Coria institute for their consistent help during my stay in France.

I am deeply in debt to my parents for their love, encouragement and support. They are ordinary farmers, but they are extraordinary in my heart because they have tried their best to support me for my study and teach me how to behave. Thanks to my wife for her persistent encouragement and support. The warm family atmosphere allows me to concentrate on the research work.

Finally, I especially thank the scholars and experts for reviewing this thesis. Qingwei Duan

2020.04.06 Xidian University (China) & Universit´e de Rouen Normandie (France)

(9)

(10)

致谢 . . . I 1 绪论 . . . 1 1.1 研究背景和意义 . . . 1 1.2 粒子光散射研究现状 . . . 3 1.2.1 解析方法. . . 3 1.2.2 数值方法. . . 4 1.2.3 GOA光散射计算方法 . . . 6 1.2.4 VCRM光散射计算方法 . . . 9 1.3 论文主要内容及框架 . . . 10 1.4 论文创新之处 . . . 11 2 圆柱、球光散射的GOA计算方法 . . . 15 2.1 概述 . . . 15 2.2 无限长圆柱的光散射 . . . 15 2.2.1 计算方法. . . 15 2.2.2 结果与讨论 . . . 23 2.3 球形粒子的光散射 . . . 27 2.3.1 计算方法. . . 27 2.3.2 结果与讨论 . . . 29 2.4 本章小结 . . . 32 3 任意光滑截面柱体光散射的VCRM计算方法 . . . 35 3.1 VCRM与GOA的区别 . . . 35 3.2 计算方法 . . . 36 3.2.1 射线追迹. . . 37 3.2.2 振幅计算. . . 39 3.2.3 相位 . . . 44 3.3 散射体描述 . . . 47 3.4 计算结果与讨论 . . . 49 3.4.1 不同形变柱体的散射特性 . . . 51 3.4.2 不同折射率下的散射特性 . . . 53 3.4.3 不同入射角下的散射特性 . . . 54 3.5 本章小结 . . . 57

(13)

4 基于VCRM的大尺寸非球形粒子三维光散射场计算方法. . . 59 4.1 三维散射强度计算所面临的挑战 . . . 59 4.2 三维射线追迹 . . . 59 4.3 波前曲率 . . . 61 4.4 交叉极化 . . . 67 4.5 三维散射中的相位计算 . . . 70 4.6 三维散射计算中振幅和相位的插值 . . . 72 4.7 本章小结 . . . 77 5 真实射流对平面波的三维散射场计算及实验验证 . . . 79 5.1 真实液体射流简介 . . . 79 5.2 射流三维光散射场的模拟结果 . . . 82 5.3 实验装置 . . . 87 5.4 模拟结果的实验对比 . . . 90 5.5 本章小结 . . . 90 6 真实射流对椭圆高斯波束的三维散射 . . . 93 6.1 椭圆高斯波束的射线模型 . . . 93 6.2 光程引起的相位变化 . . . 101 6.3 与广义米理论的对比 . . . 102 6.4 射流的椭圆高斯波束散射场. . . 105 6.5 波束发散角对散射场的影响. . . 108 6.6 三维散射场的空间分布特性. . . 113 6.7 本章小结 . . . 116 7 总结与展望 . . . 119 7.1 总结 . . . 119 7.2 展望 . . . 121 参考文献 . . . 125 作者简介 . . . 141

(14)

List of Figures

Fig 2.1 Schematic diagram of an infinite circular cylinder illuminated by a plane wave. . . 16 Fig 2.2 Ray path in the transversal section of circular cylinder. . . 16 Fig 2.3 Schematic diagram of the divergence of a ray pencil interacting with a

cir-cular cylinder. . . 18 Fig 2.4 Schematic diagram of the virtual rays (dotted) serving as the references in

calculating the optical paths of scattered rays. . . 21 Fig 2.5 Two types of focal lines defined by van de Hulst [1] in the scattering by a

sphere. For a circular cylinder concerned in this section, only the focal lines of type A exist. . . 22 Fig 2.6 Comparison of the scattered intensity calculated by GOA with that by LMT

for different size parameters. m = 1.3322. . . 24 Fig 2.7 The total and the single-order scattered intensity by a circular cylinder for

two polarizations. For clarity, only the scattered light of p = 0, 1, 2 and 3 are presented. . . 25 Fig 2.8 First-order rainbow calculated by GOA for an infinite circular cylinder. The

calculation parameters are the same with those in Fig. 2.7(a). . . 26 Fig 2.9 Comparison of the scattered intensity calculated by GOA with that by LMT

for three spherical particles of different radii. m = 1.3322. . . 30 Fig 2.10 Surface wave contributions for two spherical particles of different radii. . . 31 Fig 2.11 Zoomed view of Fig. 2.9(c) in linear scale for the first-order rainbow

calcu-lated by GOA and LMT. . . 32 Fig 3.1 An infinite cylinder of cross section f (x, y) = 0 under normal incidence.

The cylinder axis (z axis) is perpendicular to the paper plane. . . 37 Fig 3.2 Local coordinate system (ˆn, ˆτ) to describe the wave vectors at one interaction. 38 Fig 3.3 Notations for the order, the exit point and the direction of a scattered ray. . 39 Fig 3.4 Schematic diagram of the incremental variation of wavefront in

homoge-neous medium. . . 41 Fig 3.5 Variation of the wavefront of a ray pencil refracted by a circular cylinder. . 42

(15)

Fig 3.6 Schematic diagram in counting the optical path of a p = 1 ray relative to the reference ray. . . 45 Fig 3.7 The focal line for a converging wave of cylindrical wavefront. . . 47 Fig 3.8 Geometry of the composite elliptical cylinder (CEC). . . 48 Fig 3.9 Comparison of VCRM with LMT, DSE and GOA for the scattered intensity

of plane wave by a circular cylinder. The curves for DSE, LMT and GOA have been offset respectively by 10−4_{, 10}−2_{and 10}2_{for clarity. . . .} ₅₀

Fig 3.10 Comparison of the calculated scattering diagram of an elliptical cylinder (b1 = b2 = b = 50 µm, a = 1.2b) with that of a long ellipsoid (a = 60 µm,

b = 50 µm, c = 5000 µm). For clarity, diffraction is not included. . . 50 Fig 3.11 The variation of the axis ratios ξi(bi/a) with the equivalent radius r0. . . . 52

Fig 3.12 The scattered intensity of plane wave by five CECs of diverse deformations. 53 Fig 3.13 The scattering diagrams of a CEC as the temperature T goes up from 5◦_C

to 30◦_{C. The cross section of the CEC takes the shape of a raindrop of}

equivalent radius 0.75 mm. For clarity, the curves for T = 10◦_{C, 15}◦_C,...and

30◦_{C have been shifted by 10}2_{, 10}4_{,...and 10}10_{, respectively. . . .} ₅₄

Fig 3.14 Detail for the variation of the first-order rainbow as the temperature goes up. 54 Fig 3.15 Variation of the scattering diagram of a CEC as the incident angle θ0varies

from −20◦to 20◦in steps of 10◦. The cross section of the CEC is the same as the one used in Fig. 3.13. m = 1.333. . . 55 Fig 3.16 The scattering angles of the first-order, the second-order and the fifth-order

rainbows for different incident angles. (a) a circular cylinder of radius 0.75 mm; (b) a deformed CEC of equivalent radius 0.75 mm (the one used in Fig. 3.15). m = 1.333. . . 56 Fig 4.1 Local coordinate system (ˆτ, ˆn, ˆe_⊥) for describing the incident wave vector

⃗

ki, the reflected ⃗kr and the refracted ⃗kt. . . 60

Fig 4.2 Schematic diagram of a local wavefront. . . 62 Fig 4.3 Evolution of a local wavefront within a homogeneous medium. . . 62 Fig 4.4 Schematic diagram of the sudden change of wavefront caused by refraction. 63 Fig 4.5 Illustration of the two coordinate systems (ê1, ê2) and (ê(i)_∥ , ê⊥) within the

vibration plane of incident electric vector. . . 68 Fig 4.6 Definition of the base vectors (ê(i)_∥ , ê_⊥), (ê_∥(r), ê_⊥) and (ê(t)_∥ , ê_⊥) for describing

the parallel and the perpendicular components of the incident, reflected and refracted electric vectors. . . 69

(16)

Fig 4.7 Schematic diagram in calculating the optical path of a scattered ray relative to the reference ray. For clarity, an example for a scattered ray of order p =1 is shown. . . 71 Fig 4.8 Definition of the direction (φ, ψ) of a scattered ray in the 3D space. . . 72 Fig 4.9 Schematic diagram of the 3D scattering space discretized uniformly with

grid points. . . 73 Fig 4.10 Schematic diagram of two groups of scattered rays (marked as the sample

points) which are folded with each other. . . 73 Fig 4.11 Procedure of the triangulation. (1) In each iteration, only two adjacent

lay-ers of incident rays are considered; (2) The scattered rays are irregularly dis-tributed in terms of the scattering direction (φ, ψ); (3) The triangles meshed in the triangulation of the sample points. . . 74 Fig 4.12 The Delaunay criterion used in triangulation. (a) Ai+1 is not a successful

candidate since the circumcircle of triangle △AiBjAi+1contains other sample

points; (b) Bj+1with AiBj forms successfully a triangle. . . 74

Fig 4.13 Inclusion of a grid point inside a meshed triangle. The amplitude and the phase at the enclosed grid point, referred to as the query point (φq, ψq), are

to be interpolated. In the right illustrates this query scattering direction and the three sampled rays around it. . . 75 Fig 4.14 Schematic diagram of two intersecting triangles. A grid point inside the

intersecting area is an interference point, where the interference of light must be taken into account. . . 76 Fig 5.1 The image and the extracted profile of the water jet near the orifice. . . 80 Fig 5.2 The stream-wise curvature of jet surface as function of the falling height h. 81 Fig 5.3 Schematic diagram of the ray path for a light ray interacting with a liquid jet. 81 Fig 5.4 Experimental observation of the scattering field near the first-, the second-,

the fifth- and the sixth-order rainbows by a capillary water jet. . . 81 Fig 5.5 Tracing of the scattered rays: (a) by an infinite circular cylinder; (b) by a

real jet. . . 83 Fig 5.6 The configuration and the corresponding 3D scattered intensity by the liquid

jet. (a) the beam profile incident on the jet; (b) the scattering field near the first- and second-order rainbows. For clarity, the intensity of p = 0 and p =3, I0and I3, have been amplified by a factor of 10. . . 85

(17)

Fig 5.7 The scattered intensity near the first- and the second-order rainbows (φ = [125.0◦_{, 140.5}◦_{]) for an infinite cylinder and a real jet.} _{. . . .} ₈₆

Fig 5.8 The 3D scattered intensity by the water jet when h = 3.0 mm. The other parameters are the same with those in Fig. 5.7(b). . . 87 Fig 5.9 Experimental setup for measuring the scattered intensity by a liquid jet. . . 88 Fig 5.10 Schematic diagram of the experimental setup. . . 89 Fig 5.11 Experimental observation of the scattered intensity of plane wave by the

water jet near the first-order and second-order rainbows. For clarity, the intensity of the p = 0 and p = 3, I0 and I3, have been amplified by a factor

of 10. h = 2.4 mm. . . 89 Fig 5.12 Experimental examination of the simulation results for the 3D scattered

in-tensity near the first-order and second-order rainbows of a real jet. The hot maps present the relative intensity in the images captured by experiment. . 91

Fig 6.1 The coordinate system ( ˆu, ˆv, ˆw) used in describing an elliptical Gaussian beam of beam-waist radii r0uand r0v. For a circular Gaussian beam, r0u = r0v

and Ou = Ov = OG. . . 94

Fig 6.2 Variation of the intensity and the iso-phase surface of an elliptical Gaussian beam: (a) the intensity profiles at different axial positions; (b) three iso-phase surfaces sampled respectively at w = −125 µm, w = −25 µm and w = 75 µm. The vectors normal to the iso-phase surfaces depict the propagation directions. . . 96 Fig 6.3 The calculated principal directions for the saddle wavefront in Fig. 6.2(b). 99 Fig 6.4 Two coordinate systems established respectively for the particle and the beam. 99 Fig 6.5 Schematic diagram for calculating the optical path of a scattered ray relative

to the reference ray (dotted blue line) in the 3D scattering of a shaped beam. 101 Fig 6.6 Configuration in calculating the scattering of circular Gaussian beam by a

spherical particle. . . 103 Fig 6.7 Comparison of VCRM with GLMT for the scattered intensity of circular

Gaussian beam by a spherical particle of different refractive indices m. For clarity, the intensity by VCRM has been shifted by a factor of 10. . . 104

(18)

Fig 6.8 Simulated result of the 3D scattered intensity of elliptical Gaussian beam by a real jet: (a) the profile of the incident elliptical Gaussian beam on the jet; (b) the scattering field near the first- and second-order rainbows. For clarity, the intensity of the p = 0 and p = 3, I0 and I3, have been amplified

by a factor of 10. . . 106

Fig 6.9 The scattering field near the first- and second-order rainbows when the in-cident position moves down to h = 3.5 mm. . . 108

Fig 6.10 Simulated result of the 3D scattered intensity of a circular Gaussian beam with beam-waist radius comparable to the jet radius. (a) the profile of the incident circular Gaussian beam on the jet; (b) the scattering field near the first- and second-order rainbows. For clarity, the intensity of p = 0 and p =3, I0and I3, have been amplified by a factor of 10. . . 109

Fig 6.11 An elliptical Gaussian beam with a considerable divergence angle in the vertical xz plane: (a) the beam diameter in the vertical xz plane; (b) the intensity profiles in the transversal yz planes sampled at three positions. . . 110

Fig 6.12 Single-order scattered intensity of the tightly focused elliptical Gaussian beam modeled in Fig. 6.11 by the liquid jet. For clarity, I0and I3have been amplified by a factor of 10. . . 111

Fig 6.13 Simulated 3D scattered intensity for two elliptical Gaussian beams of dif-ferent divergence angles in xz plane. . . 112

Fig 6.14 Zoomed view of the interference fringes near the first rainbow. . . 114

Fig 6.15 Single-order intensity of the p = 0 and p = 2 scattered light along the sampled line (A). . . 114

Fig 6.16 The rainbow fringes sampled at four different elevation angles. . . 115

Fig 7.1 The shapes of natural raindrops with increasing radii (from [6]). . . 121

Fig 7.2 A wavy sea surface when the wind speed is 4.4 m/s. . . 122 Fig 7.3 Preliminary result for the temporal-spatial distribution of the light rays

trans-mitted from the sea at the height of a satellite (485 km). The colormap illus-trates the received photon numbers (relative values). The wind speeds for the left column and the right column are 4.4 m/s and 12.3 m/s, respectively. 123

(19)

(20)

List of Tables

Table 3.1 Geometric parameters of the CEC at different equivalent radius r0 (unit:

mm) . . . 52

Table 3.2 Refractive indices of pure water for λ = 0.6328µm at different temperatures T (◦_{C) . . . .} ₅₃

Table 6.1 Parameters of the elliptical Gaussian beam in Fig. 6.2. . . 95

Table 6.2 Parameters of the circular Gaussian beam in Fig. 6.7. . . 103

Table 6.3 Parameters of the elliptical Gaussian beam in Fig. 6.8. . . 105

Table 6.4 Parameters of the circular Gaussian beam in Fig. 6.10. . . 108

Table 6.5 Parameters of the tightly focused elliptical Gaussian beam in Fig. 6.11. . . 110

(21)

(22)

Annotation of Symbols

Greek Symbols

γ : divergence angle of incident beam

ε : variation of amplitude due to reflection and refraction coefficients ζ : curvature of wavefront

η : distance factor

θ : scattering angle in 2D scattering Θ : projection matrix

λ : wavelength of light

ρ : curvature radius of particle surface Σ_inc : incident plane

ˆτ : unit vector tangent to the particle surface

ϕ : phase

φ : azimuth angle in 3D scattering ψ : elevation angle in 3D scattering

Subscripts or superscripts

X : _{⊥ or ∥ respectively for perpendicular or parallel polarization} ⊥ : the component of electric field perpendicular to the incident plane ∥ : the component of electric field parallel to the incident plane p : for p-order scattered ray

(i) : parameters for incident ray (r) : parameters for reflected ray (t) : parameters for refracted ray n : normal component

(23)

Roman Symbols

a : radius of particle

C : curvature matrix of particle surface D : divergence factor

⃗

E : electric vector

h : distance from the exit of orifice H : Hessian matrix

Jr, Jt : Jones matrices respectively for reflection and refraction

k : wave number

⃗k : wave vector

kn : normal component of wave vector

kτ : tangent component of wave vector

m : relative refractive index of particle

Mt : refractive index of the refracted medium relative to the incident medium

M : coordinate transformation matrix

ˆn : unit vector normal to the particle surface

Ou, Ov : positions of the beam waists respectively in ( ˆw,ˆu) and ( ˆw,ˆv) planes

p : the order of scattered ray Q : curvature matrix of wavefront

r : geometric length from the exit point to the observation point r0u, r0v : beam-waist radii respectively in ( ˆw,ˆu) and ( ˆw,ˆv) planes

r0y, r0z : beam-waist radii respectively in ( ˆx, ˆy) and ( ˆx, ˆz) planes

r_⊥, r_∥ : Fresnel reflection coefficients R : curvature radius of wavefront

s : geometric length between two successive interactions t_⊥, t_∥ : Fresnel refraction coefficients

ˆu1, ˆu2 : principal directions of wavefront

ˆv1, ˆv2 : principal directions of particle surface

Wp : exit point of a p-order scattered ray

( ˆu, ˆv, ˆw) : coordinate system for elliptical Gaussian beam ( ˆx, ˆy, ˆz) : Cartesian coordinate system

(24)

Abbreviations

CEC : Composite Elliptical Cylinder : 组合椭柱

DDA : Discrete Dipole Approximation : 离散偶极子近似

DSE : Debye Series Expansion : 德拜级数展开

EBCM : Extended Boundary Condition Method : 拓展边界条件方法 FDTD : Finite Difference Time Domain : 时域有限差分

GLMT : Generalized Lorenz-Mie Theory : 广义洛伦兹-米理论

GOA : Geometrical Optics Approximation : 几何光学近似

IIM : Invariant Imbedding Method : 不变嵌入法

LMT : Lorenz-Mie Theory : 洛伦兹-米理论

MLFMA : Multilevel Fast Multipole Algorithm : 多层快速多极子算法

PO : Physical Optics : 物理光学

POA : Physical Optics Approximation : 物理光学近似方法

(25)

(26)

Chapter 1 Introduction

This chapter introduces the research background and significance of the thesis from the aspect of particle scattering. It reviews the research history and current status of related fields, briefly describes the analytical and numerical methods for light scattering, and puts more effort on the introduction of two approximation methods: the geometrical optics ap-proximation (GOA) and the vectorial complex ray model (VCRM), where the incident wave is represented by a collection of discrete rays. Finally, the research content and method, the content of each chapter, the overall framework, the results and the innovations of the thesis are introduced.

1.1 Research background and significance

Light scattering by particles is a field of active research with high relevance for science and engineering. It promotes our understanding of the interaction between light and particles and concerns an increasing number of modern technologies, ranging from the measurement and the manipulation of particles to optical communications in turbid media.

Particles are abundant in natural and artificial environments. In many fields of science and engineering, such as combustion diagnosis, atmospheric optics, remote sensing, scatter-ing by interplanetary dust grains, bio-optical imagscatter-ing, colloidal chemistry, materials science, coating technology and particle sizing technology, one often has to deal with particles rang-ing from nanoscale to macroscopic.

Since the advent of laser, optical measurement technologies have been favored for its non-contact, accurate and fast merits. The techniques based on elastic light scattering for particle analysis (such as phase Doppler, extinction method, rainbow technology and small particle imaging technology) have found wide applications. The basic principle of these techniques is based on the fact that the intensity distribution, polarization characteristics and spectral characteristics of scattered light are all related to the particles [1, 2]. According to the characteristics of the scattered light, a lot of information about the particles can be retrieved, such as the size, shape, speed of motion, temperature, density and compositions. Efficient theoretical and numerical tools to predict the scattering characteristics of light by various particles are thus essential.

(27)

particles. Each method generally has its own range of applicability determined primarily by the particle size relative to the wavelength of incident light. The scattering by those particles that are very small compared to the wavelength can be calculated by the Rayleigh approximation [1, 2]. The particles of size comparable to the wavelength lie in the range commonly called the resonance region, where the separation of variables techniques[1, 2] and numerical methods [3, 4] are usually utilized. And those particles of size much larger than the wavelength can be addressed by the approximation methods, in which the incident wave is represented as a collection of discrete rays, for example, the GOA method [1].

Among the particles of various forms in nature, one group of them are of smooth surface due to surface tension, such as the raindrops [5–8], droplets in industry [9–13], underwater bubbles [14–29], liquid jets [30–36] and so on. As the effects of gravity and air forces become non-ignorable, their geometric shapes are deformed from a sphere or an infinite cylinder, indicating that their scattering characteristics cannot be resolved by the separation of variables techniques. Besides, their size parameters1 _{usually range from hundreds to}

thousands or even larger, making the calculation far beyond the capabilities of the existing numerical methods. The developments of alternative approaches allowing fast and accurate (at least asymptotical) calculations are needed for carrying out in-situ characterization of these particles.

For a nonspherical particle of size much larger than the wavelength (size parameter is generally over 100), the approximate methods based on the assumption that the light wave can be represented by a bundle of discrete rays provide feasible ways for calculating the scat-tering field. Many researchers have contributed to the development of GOA for dealing with spherical or spheroidal particles/bubbles [1, 37–43], faceted particles [44, 45] and natural raindrops [6]. In a GOA model, each ray is characterized by its direction, amplitude, polar-ization and phase. However, for a particle of curved surface, the GOA encounters difficulties or even obstacles in calculating the shift in ray intensity due to the divergence/convergence of wave (divergence factor) and the shift in phase due to focal lines, except for a spherical or spheroidal particle. For this reason, the applications of GOA are under restrictions.

In this context, Ren et al. [46] proposed the vectorial complex ray model (VCRM), a new model for describing the interaction of light rays with a large particle. The distinctive features of VCRM are that: 1) the curvature of wavefront is integrated as an intrinsic prop-erty of a light ray, by which one can directly calculate the divergence factor and the phase 1_{Size parameter depicts the size of particle compared to the wavelength of light. It is calculated as 2πa/λ with a being} the equivalent radius of particle and λ being the wavelength of light.

(28)

shift due to focal lines; 2) the ray directions and the Fresnel coefficients are calculated by the wave vectors and their components; 3) the interference of all scattered rays in the whole region is taken into account.

Since the light scattering by a nonspherical particle of size much larger than the wave-length is still a challenging problem in the field of light scattering, this thesis is devoted to the extension of VCRM, with aims to solve the light scattering by infinite cylinders of vari-ous forms, and more importantly, to solve the scattered intensity of plane wave and shaped beam by large nonspherical particles in 3D space.

1.2 Research status of light scattering by particles

This section summarizes the research status of various calculation methods for the light scattering by particles.

1.2.1 Analytical methods

The search for an exact analytical solution to the scattering field has been traditionally reduced to solving the vector Helmholtz equation for the time-harmonic electric field using the separation of variables techniques in one of the few coordinate systems in which this equation is separable. For this reason, the separation of variables technique are limited to particles of simple forms such as spheres, spheroids, circular or elliptical cylinders.

Lorenz [47] and Mie [48], independently, derived the solution for an isotropic homo-geneous sphere in 1890 and 1908. The method that they proposed is now the well-known Lorenz-Mie theory (LMT). The LMT is a rigorous solution of the Maxwell equations and contains all the effects that contribute to the scattering. It was expounded later in depth by Van de Hulst [1] and by Bohren and Huffman [2]. Aden and Kerker provided a solution for the scattering by a coated sphere in 1951 [49]. Wait in 1955 presented a solution for the scat-tering by a homogeneous, isotropic, infinite circular cylinder [50]; and it was extended to the scattering by an isotropic, infinite elliptical cylinder by Kim and Yeh in 1991 [51]. Oguchi in 1973 and Asano and Yamamoto in 1975 derived a general solution for the scattering by homogeneous, isotropic spheroidal particles [52, 53].

To give a clear interpretation to the various physical processes in scattering, the Debye series expansion (DSE) method [54] was developed for a homogeneous [55] and coated [56] sphere respectively in 1992 and 1994. In 2006, Li et al. [57] derived the formula for the DSE in the light scattering by a multilayered sphere and introduced an efficient algorithm permitting stable calculation for a large multilayered sphere. Wu and Li in 2008 [58]

(29)

pre-sented a simplified but rigorous iterative formula for the DSE in the light scattering by an infinite circular cylinder of multiple layers. In 2010, Shen and Wang [59] provided a stable, reliable and robust algorithm for the DSE calculation of the scattering of plane wave by a uniform sphere in a wide range of sizes and refractive indices. Xu and Lock contributed to the development of DSE for the light scattering by a spheroid [60], and by a homogeneous or coated nonspherical particle in combination with the extended boundary condition method [61, 62].

With the advent of lasers and their growing applications, the LMT met one of its fun-damental limitations, i.e., the assumption that the incident wave must be a plane wave. To address this problem of LMT, in 1988, G. Gouesbet and G. Gréhan et al. presented a theo-retical description of the scattering of a Gaussian beam by a homogeneous and isotropic par-ticle of spherical shape, which is now known as the generalized Lorenz-Mie theory (GLMT) [63, 64]. An equivalent approach for the scattering by a dielectric sphere located arbitrarily within a Gaussian beam was presented by Barton et al. [65, 66]. In the framework of GLMT, Ren et al. in 1994 and 1996 derived the radiation pressure (pushing force) and the reverse radiation pressure (pulling force) exerted by Gaussian beam on spherical particles [67, 68]. Onofri et al. [69] presented a solution in the framework of GLMT for the scattering of an arbitrary shaped beam by a multilayered sphere. Ren and Méès et al. [70, 71] extended the GLMT to the scattering of Gassian beam by infinite circular cylinders, while Gouesbet and Méès contributed to the extension of GLMT to the scattering by infinite elliptical cylinders [72, 73]. Han and Wu et al. devoted to the scattering of shaped beams by spheroidal parti-cles [74–76]. Wang and Han et al. applied the GLMT to the scattering of shaped beam by a sphere with an eccentrically located spherical inclusion [77, 78]. Later, Wang et al. pro-posed an algorithm in the framework of GLMT to address the internal field distribution of a radially inhomogeneous droplet illuminated by an arbitrary shaped beam in a wide range of size parameters [79]. More comprehensive reviews of the GLMT can be found in [80, 81].

It is unlikely that these analytical solutions will be significantly extended in the future. Indeed, the solution for the simplest finite nonspherical particles, spheroids, is already so complex that it behaves like a numerical solution as Mishchenko pointed out in [3].

1.2.2 Numerical methods

Numerical methods provide exact solutions to the scattering of electromagnetic/light wave by nonspherical particles. Most of them fall into two broad categories [3]: the dif-ferential equation methods which compute the scattering field by solving the vector wave

(30)

equation in the frequency or in the time domain; and the integral equation methods which are based on the volume or surface integral counterparts of Maxwell’s equations, with ex-ceptions of those hybrid techniques or methods. Here a brief introduction is made to those commonly used numerical methods for the scattering of light by nonspherical particles.

In the discrete dipole approximation (DDA), a scatterer is divided into small cubical subvolumes (“dipoles”), and the interaction between the dipoles are approximated based on the integral equation for the electric field [82–85]. It can be applied to simulating the light scattering by finite 3D objects of arbitrary geometry.

The finite-difference time domain (FDTD) method solves the Maxwell equations in the time domain by using the finite-difference analog. It was originally developed by Yee in 1966 [86]. After that, the FDTD method has been extensively applied to solving various electromagnetic problems [87–94]. The most relevant literatures are [44, 95], where Yang and Liou et al. proposed the solutions for the light scattering by ice crystal particles.

The T-matrix method may be the most accurate and efficient numerical method for solv-ing the scattersolv-ing of electromagnetic wave by a nonspherical particle, and can achieve the widest range of size parameter (up to 200 or 300 dependent on the particle shape [96–99]). As one renowned method to compute the T-matrix in the T-matrix formulation of light scat-tering, the extended boundary condition method (EBCM), or Waterman’s T-matrix method, was initially proposed by Waterman in 1965 and 1971 [100, 101]. It was later developed by Barber and Hill [102], Mishchenko [103–105] and others. The standard EBCM [105] is very efficient but will encounter a loss of precision when the particle size becomes larger, the maximum size being affected by the particle aspect ratio. On the other hand, the invariant imbedding method (IIM) was originally proposed by Johnson in 1988 [106]. Bi, Sun and Yang et al. [107, 108] and others have made significant contributions to the development of IIM. It is based on an electromagnetic volume integral equation and obtains the T-matrix by growing the scattering volume incrementally in a shell-by-shell manner. The IIM is applica-ble to particles of relatively large size parameters and extreme aspect ratios. But it is not as efficient as the EBCM due to the large number of differential shells required to discretize the particle volume. Bi and Yang et al. [96] improved the computational efficiency of IIM by combining the IIM with the LMT and applied to the spheroids and cylinders of size param-eters beyond the convergence limit of EBCM. They also proposed a numerical combination of IIM with EBCM [97], which not only enhanced the efficiency of IIM but also extended the convergence limit of EBCM.

(31)

speak-ing, when the size parameter of particle is over 100, the computation using a numerical method becomes rather time- and memory-consuming.

1.2.3 GOA

The geometric optics approximation (GOA) (otherwise known as the ray tracing or ray optics approximation) is an approximate method for the light scattering by large particles, where the incident wave is described by a bundle of light rays. Van de Hulst in his renowned book [1] presented a relatively systematic description of GOA for the scattering of plane wave by a spherical particle, where the direction, amplitude and phase for each scattered ray of perpendicular or parallel polarization were derived. Besides, many researchers have extended and applied the GOA to various scattering problems including:

• The scattering by particles of various forms. Glantschnig and Chen [109] simplified the calculation for the superposition of the externally reflected rays and the refracted rays with the diffraction and obtained a closed formula for the scattering in the forward angu-lar range (0◦ _{− 60}◦_{) by a water droplet. Adler and Lock et al. [110, 111] examined the}

scattering by an infinite cylinder of deformed cross section in terms of ray tracing, rainbow angles and the rainbow intensity due to Fresnel coefficients. Hovenac [112] developed an algorithm for predicting the far-field scattering from a particle being symmetric about the optical axis. Lock [37, 38] addressed the reflection, transmission and diffraction of the light rays scattered by an arbitrarily oriented spheroid. Sadeghi et al. [6] extended the GOA to the 3D scattering field near the rainbows produced by natural large raindrops (neither spheres nor spheroids). Yang and Liou [45] proposed a new geometrical optic-s model which uoptic-sed the ray-tracing technique to optic-solve the near field on the ice cryoptic-stal surface and then transformed the near field to the far field based on the electromagnet-ic equivalence theorem. Bi et al. [113] assessed the uncertainties with the conventional geometrical optics in remote sensing and radiative transfer simulations.

• The scattering by bubbles. In 2008, Yu et al. [41] studied the scattering of plane wave by a spherical bubble based on GOA, where the total reflection effect was taken into account to improve the calculation accuracy. In 2012, He et al. [42] studied the scattering of plane wave by a spheroidal bubble with end-on incidence, where the effects of the size and the aspect ratio of bubble and the width of incident beam on the scattering patterns were analyzed. In 2016, Sentis and Onofri et al. [114] improved the GOA by combining it with the physical optics approximation (POA) in modeling the scattering properties of large spherical bubbles, where the interference between higher-order rays, the Goos–H¨anchen

(32)

shift, the tunneling phase and the weak caustic associated with the critical angle were taken into account.

• The scattering by coated particles. Lock et al. [56] presented an intuitive interpretation to the first-order rainbow from a coated sphere by using ray theory in 1994. In 2004, Xu et al. [115] provided an algorithm by means of GOA for calculating the scattered intensity by a coated spherical particle in the forward angular range (0◦_{− 60}◦). In 2014, Zhai et al. [116] examined the scattering processes of a coated sphere with the GOA method. They parameterized the light rays interacting with a coated sphere and simplified the calculation for those terms of degeneracy paths and repeated paths.

• Ray tracing in inhomogeneous media. The path of a light ray in inhomogeneous media is curved. In the 1960s, Montagnino [117] and Marchand [118] contributed to the de-velopment for the methods for tracing the ray paths in inhomogeneous media. In 1982, Sharma et al. [119] proposed an efficient method for tracing the light rays passing through gradient-index media. Later, Sharma achieved a method for computing the optical path in gradient-index media, which proved to be more accurate and faster after a compari-son with other methods [120]. As for the spherical particles of inhomogeneous media, Li et al. [121] presented a solution in 2007 to the scattered intensity of plane wave by a gradient-index sphere in the forward angular range (0◦_{− 60}◦_{). In 2008 and 2015, Lock et}

al. [122, 123] analyzed the scattering of plane wave by a Luneburg lens.

• The scattering by chiral particles. In 2015, the rainbow angles of a chiral sphere were calculated using ray tracing by Wu et al. [124]. In 2019, the scattering pattern of plane wave by a chiral sphere in the forward direction (0◦_{− 90}◦_{) has been solved with GOA by}

Lu et al. [125].

• The scattering of shaped beam. In 2006, Xu et al. extended the GOA for calculating the scattered intensity of circular Gaussian beam by a sphere [39] and by a spheroid with end-on incidence [40], where the Gaussian beam was described by a bundle of light rays of different propagation directions, amplitudes and phases. To calculate the optical forces of an arbitrary shaped beam, Shao et al. in 2019 decomposed an arbitrary light beam into plane waves. The intensity of each plane wave was dependent on the Fourier angular spectrum; and each plane wave was further represented by a bundle of light rays parallel in propagation direction [126].

(33)

to characterize the plane wave propagating in a lossy media. Yu et al. in 2009 presented a method based on GOA to address the light scattering by an absorbing sphere, where the effective refractive index and the effective refractive angle were introduced, and the formulas for the phase shifts due to reflections and refractions were derived [128]. In 2018, Lindqvist et al. [129] derived a ray-optics solution which took into account the inhomogeneous nature of the represented wave inside an absorbing particle and applied it to the light scattering by ice particles in the near-infrared wavelengths.

• The coupling with physical optics. In 2015, Huang et al. combined the GOA and the concept of divergence factor with the physical optics (PO) and the physical theory of diffraction for the scattering by perfectly electrical conducting targets [130]. Sentis et al. in 2016 coupled the GOA with the POA and developed an improved version of GOA, which allows one to predict the scattering pattern by large bubbles with high accuracy [114].

• The application to optical forces. Ashkin in 1992 calculated the forces of single-beam gradient radiation pressure on micron-sized dielectric spheres in the ray optics regime [131]. Zhou et al. in 2012 calculated the optical forces on a triaxial ellipsoid by vectorial ray tracing [132]. Later Shao et al. in 2019 combined the Fourier optics and ray optics for calculating the optical force of an arbitrary shaped beam exerted on a spherical particle [126].

• The application to inelastic scattering. The theoretical treatment of inelastic scattering by wave theory is rather complicated and, most of the calculations are limited to a spherical particle of homogeneous medium or with inclusions of size smaller than the wavelength. Being flexible and efficient in computation, the ray optics was utilized to calculate the inelastic scattering by a large spherical particle [133, 134], by a particle of any shape where the particle surface was described by triangles [135], and by a spherical particle with inclusions [136].

A GOA method provides clear insight into the reflection and refraction processes and has advantage in the cases where the exact numerical methods are hard to achieve or no rigorous theory exists, as concluded by Xu et al. [39]. However, a careful reader may notice that most of the GOA methods available today have addressed only the scattered intensity by spherical or spheroidal particles, or faceted ice crystals. This is mainly due to the fact that although the GOA is flexible in principle, it usually encounters difficulties or even obstacles

(34)

in accounting for the divergence factor and the phase shift due to focal lines, which restricts the precision and application of GOA.

1.2.4 VCRM

To resolve these difficulties encountered in GOA, Ren et al. in 2011 proposed the vec-torial complex ray model (VCRM) [46]. In VCRM, the curvature of wavefront is integrated as an intrinsic property of a light ray, by which the divergence factor and the phase shift due to focal lines can be calculated directly. Furthermore, the ray directions and the Fresnel coefficients are calculated by the wave vectors and their components, which avoids the te-dious calculation when a nonspherical particle is involved. Although it is a high-frequency approximation model, the VCRM provides one feasible way for calculating the scattered intensityof plane wave or shaped beam by a large particle of arbitrarily smooth surface2_.

The validity of VCRM has been examined numerically by comparing with the multi-level fast multipole algorithm (MLFMA) for the light scattering by a large ellipsoidal par-ticle [137]. Furthermore, Onofri et al. experimentally examined the VCRM for the light scattering by a large oblate droplet trapped in an acoustic field [138].

Ren et al. applied the VCRM to the light scattering in the transversal plane of an ellipsoidal particle in 2012 [139]. Jiang et al. applied it to the scattering of plane wave and Gaussian beam by elliptical cylinders in 2012 and 2013 [140, 141]. Onofri et al. retrieved the evolution of the principal curvature radii and the refractive index of an oblate droplet with a minimization method that involves VCRM predictions and experimental light scattering patterns in 2015 [138]. Sun et al. improved the three-dimensional (3D) ray tracing by considering the wave-front distortion and phase shift in the scattering by a spheroid in 2016 [142].

However, the VCRM is still in its early stage of development. The existing numerical implementations of VCRM address only the scattering in a few simple configurations. One of them is that all light rays should propagate in a two-dimensional (2D) plane, for example, in one of the symmetric planes of a spheroidal particle.

In the past few years, several researchers have been trying to find a solution to the 3D scattered intensity by large nonspherical particles in the framework of VCRM. Recently, in Yang’s doctoral thesis which was defended in the December of 2019 [143], an alternative implementation of VCRM using statistic approach has been achieved for the 3D scattered intensity by a pendent droplet. But, the statistic approach requires a huge number of photons,

(35)

at least 108_{to 10}10_{to obtain an acceptable result for a pendent droplet of 2 mm. This statistic}

approach, though effective, is so time- and memory-consuming that it violates the efficient merit of VCRM.

1.3 Main content and frame arrangement

Because of the flexible and efficient merits of VCRM, this thesis is devoted to the extension of VCRM to the light scattering by an infinite cylinder of arbitrarily smooth cross section, and more importantly, to the scattered intensity of plane wave and shaped beam by a large nonspherical particle in 3D space. This thesis focuses mainly on the following issues: 1. The extension of VCRM to the 2D scattering of plane wave by an infinite cylinder of any

smooth cross section.

2. The calculation method based on VCRM for the 3D scattered intensity by a large particle of any smooth surface.

3. The application of the proposed method to calculating and analyzing the scattered inten-sity of plane wave by a real liquid jet of complex surface in 3D space.

4. The examination of the proposed method by experiments.

5. The extension to the 3D scattered intensity of shaped beam, an elliptical Gaussian beam for example.

The scattering by inhomogeneous particles is not involved here, because the variation of wavefront in inhomogeneous media remains unknown.

The thesis includes seven Chapters. The content of each Chapter is outlined as follows: Chapter 1 introduces the research background and the significance of this thesis, re-views the research history of related fields, points out the issues that this thesis aims to solve, introduces the main content and the frame arrangement, and outlines the research results and innovations.

Chapter 2 discusses in depth the classical GOA method for the scattering of plane wave by circular cylinders and spheres.

Chapter 3 introduces the interaction of light rays with a cylinder in the framework of VCRM and, extends the VCRM to the scattering of plane wave by an infinite cylinder of any

(36)

smooth cross section. The proposed algorithm is applied to solving the full diagram of the scattered intensity by a cylinder whose cross section takes the shape of a natural raindrop. The effects of shape deformation, refractive index and the direction of incident wave on the scattering patterns are investigated and quantitatively analyzed.

Chapter 4 proposes the calculation method for the 3D scattering of plane wave by a large particle of any smooth surface. In the framework of VCRM, the ray tracing, divergence factor, phase shifts due to focal lines and optical path, and cross polarization in 3D scattering are addressed by an elegant way using vectorial rays and wave-front curvature. To account for the superposition of scattered ray in 3D space, a triangulation-based interpolation method is proposed, thus breaking through the bottle-neck problem for VCRM in calculating the 3D scattered intensity.

Chapter 5 realizes the calculation for the 3D scattered intensity of plane wave by a real liquid jet. The geometric model of a continuous water jet near the nozzle is established through image edge detection and data fitting. By applying the calculation method proposed in Chapter 4, the 3D scattering field by the jet is successfully simulated. Emphasis is put on the 3D intensity distribution near the first-order and second-order rainbows. The effect of the stream-wise curvature of jet surface on the scattering characteristics is analyzed. The difference from the scattering field by an infinite cylinder is discussed. An experimental setup is also established for measuring the 3D scattered intensity by a liquid jet, and the result by simulation is examined by the experiment.

Chapter 6 proposes a ray description method in the framework of VCRM for incident elliptical Gaussian beams. After the validation of the proposed model by comparing with the GLMT for spherical particles, the 3D scattered intensity of elliptical Gaussian beam by a real liquid jet which has a complex geometry has been successfully calculated. The scattering characteristics of the elliptical Gaussian beams of different divergence angles are investigated. The spatial characteristics of the scattering field of a tightly focused elliptical Gaussian beam is analyzed.

Chapter 7 concludes the work of the current thesis and gives perspectives for the future.

1.4 The innovations

This thesis addresses the 2D and 3D scattered intensity by a large particle of arbitrarily smooth surface in the framework of VCRM. The innovations are as follows:

(37)

infi-nite cylinder of arbitrarily smooth surface in the high-frequency limit. For the scattering by an infinite cylinder, the rigorous LMT is limited to the cylinders which have circular or elliptical cross sections; while the GOA method, except for a circular or elliptical cylin-der, is hard to calculate the divergence factor and the phase shift due to focal lines. The algorithm proposed in this thesis allows one to obtain the direction, amplitude, phase, po-larization and divergence factor for each scattered ray, and the final interference intensity of the scattered rays from an infinite cylinder of arbitrarily smooth surface. It provides an effective tool for calculating and analyzing the scattering patterns by a cylinder of cross section ranging from simple to complex.

2. The bottle-neck problem of VCRM in calculating the 3D scattered intensity has been solved. Inspired by the idea of triangulation in the area of Computer Graphics, the di-rection of each 3D scattered ray is represented as a vertex in the (φ, ψ) space, where φ and ψ are respectively the azimuth angle and the elevation angle. These vertexes which represent the directions of scattered rays are firstly meshed by triangles. Then, inside each generated triangle, interpolation is carried out for the amplitudes and phases at the grid points which are enclosed by the triangle, according to the amplitudes and phas-es of the three vertexphas-es (three scattered rays). The overlapping trianglphas-es at a grid point (φi, ψj) account for the interference effect in the scattering direction of azimuth angle φi

and elevation angle ψj. By introducing this triangulation-based interpolation method, the

calculation for the superposition of the scattered rays in 3D space can be achieved. Be-sides, the ray tracing, divergence factor, phase shifts due to focal lines and optical path, and cross polarization encountered in 3D scattering are addressed in the framework of VCRM. It provides an effective and efficient approach to the scattered intensity in 3D space by a large smooth particle of any shape.

3. The scattered intensity of plane wave in 3D space by a real liquid jet has been successfully simulated and experimentally validated. The study of the light scattering by a liquid jet is motivated primarily by the needs for developing optical means to characterize the jet size and refractive index (temperature). However, the scattering pattern of a real jet can hardly be calculated using the existing analytical theories or numerical methods because of the jet’s complex geometry and large size (compared to wavelength). In this thesis, the 3D far-field scattered intensity of plane wave by a real liquid jet has been solved. It is found that due to the stream-wise curvature of jet surface, the scattered rays of different orders by a real jet are naturally separated in the 3D space, leading to scattering

(38)

patterns that have rarely been observed. Besides, an experiment has also been carried out. The agreement between the simulated 3D scattered intensity and the experimental result validates the proposed algorithm which is based on VCRM while allows to calculate 3D scattering field.

4. Many researchers have contributed to the development and application of the GLMT for the scattering of a Gaussian beam by particles. However, the GLMT is applicable only to particles of simple forms so that the separation of variable method could be carried out. In this thesis, the ray model for incident elliptical or circular Gaussian beam is proposed in the framework of VCRM, thus making it possible to calculate the scattered intensity of laser beam by a large particle of any smooth surface. As an example, the 3D far-field scattered intensity of elliptical Gaussian beam by a real liquid jet has been achieved. The scattering characteristics near the first-order and second-order rainbows for incident elliptical Gaussian beams of different divergence angles are investigated in 3D space.

(39)

(40)

Chapter 2 Light scattering by a circular cylinder or a sphere based

on GOA

2.1 Overview

The scattering of plane wave by an infinite circular cylinder under normal incidence or by a spherical particle is one of the basic problems in the filed of light scattering [1]. When the transversal size of cylinder or the radius of sphere is much larger than the wavelength of light, the geometrical optics approximation (GOA) method provides an asymptotic solution to the scattering field. Although it is not as exact as the LMT or DSE, the calculation by GOA is much simpler and more efficient. In this chapter, the GOA method for calculating the scattered intensity by a circular cylinder under normal incidence and by a sphere are discussed. It provides the fundamental conceptions of the interaction of light rays with a particle, thus facilitating the understanding of the VCRM and its extensions in the following chapters.

2.2 Light scattering by an infinite circular cylinder

2.2.1 Calculation method

Consider an infinite circular cylinder illuminated by a plane wave as shown in Fig. 2.1. The Cartesian coordinate system Oxyz is set such that the z axis is along the cylinder axis and the x axis coincides with the direction of the incident plane wave. The transversal radius of the cylinder is denoted as a. The refractive indices of the surrounding medium and the cylinder are 1 and m, respectively. The scattering angle θ denotes the direction of a scattered ray and is counted in the xy plane from the x axis.

The incident plane wave is regarded as a bundle of light rays of same direction, equal amplitude and equal phase. For an incident light ray, it is subjected to continual reflections and refractions by the cylinder surface. Thus, theoretically speaking, there is an infinite series of emergent rays. An externally-reflected ray is of order p = 0, a transmitted ray without internal reflection is of p = 1, while an emergent ray which undergoes p − 1 times of internal reflections is a p-order scattered ray as illustrated in Fig. 2.2. Such a definition for the order of a scattered ray gives certain convenience in later calculation.

(41)

O

z

O

z

x

y

q

m Sketch of incident rays Direction of a scattered ray

Fig 2.1 Schematic diagram of an infinite circular cylinder illuminated by a plane wave.

t

b

a

_t

_'

0 p =

'

t

1 p =

'

t

2 p =

x

y

m

a

(42)

and the refracted angle β are introduced as τ and τ′_{, respectively. τ = 0 indicates the grazing}

incidence while τ = π/2 indicates the central incidence. For one of the emergent rays that originate from the same incident ray, its angle with the cylinder surface is always τ. For any light ray inside the cylinder which originates from this incident ray, its angle with the cylinder surface is a constant of value τ′_{as shown in Fig. 2.2.}

From Fig. 2.2, one can derive that the total deviation angle of a p-order emergent ray from the incident direction (x axis) is

θ′ ₌_{2τ − 2pτ}′ _(2-1)

In practice, the scattered light is observed in [0, 2π], or in the region of [0, π] considering the symmetry of the scattering field. The scattering angle θ in the interval [0, π] is related to the total deviation angle by

θ′ = c2π + qθ (2-2)

where c is an integer equal to the times that the emergent ray has crossed the x axis. q is equal to +1 or −1 ensuring that θ is in the interval [0, π].

At the first reflection, the ratio of the reflected amplitude to the incident amplitude is calculated according to the Fresnel reflection coefficients:

r_⊥= sin τ − m sin τ ′ sin τ + m sin τ′ (2-3) r_∥= msin τ − sin τ ′ msin τ + sin τ′ (2-4)

where the subscript ⊥ indicates the polarization perpendicular to the scattering plane xy, while ∥ indicates the polarization parallel to the scattering plane xy. These two polarizations are respectively along ˆe_⊥and ˆe_∥as shown in Fig. 2.1.

At a reflection inside the circular cylinder, τ and τ′ _{are reversed compared to the first}

reflection. Thus, the reflection coefficients at an internal reflection are −r⊥and −r∥.

Accord-ing to the conservation of energy, the refraction coefficient tX and the reflection coefficient

rX are related by

tX2

msin τ′

sin τ + rX

2 ₌₁ _(2-5)

where X =⊥ or ∥. Thus, the refraction coefficient (always positive) can be expressed as

tX = ( 1 − rX2 )1/2( sin τ msin τ′ )1/2 (2-6)

(43)

For a scattered ray of order p ≥ 1, it has undergone p − 1 times of internal reflections and two times of refractions: one refraction from air to cylinder and one refraction from cylinder to air. The refraction coefficients at the two refractions are denoted as tXand t′X, respectively.

According to Eq. (2-6) and the fact that τ and τ′are reversed at the two refractions, we have

tXt′X =

(

1 − rX2

)

(2-7) Then, the variation of the amplitude of a p-order scattered ray due to the reflection and refraction coefficients is obtained as

εX,p =

 

r_{(1 − r}X X2)(−rX)p−1 ifif p =p ≥ 10

(2-8)

where the term (1 − rX2) corresponds to the two refractions, while (−rX)p−1the p − 1 times

of internal reflection. Ray pencil y ds dt a t d t + t dl t q dq x 0 I I r ˆx ' ds 0 s

Fig 2.3 Schematic diagram of the divergence of a ray pencil interacting with a circular cylinder.

On the other hand, a reflected/refracted ray pencil might be converging or diverging and its intensity will be higher or lower accordingly. Consider a ray pencil of width dl in the xyplane, thickness dz along the axis of cylinder and uniform light intensity I0. It illuminates

the surface of cylinder in an area of dsdz where the complementary angle of incident angle varies from τ to τ + dτ. The incident energy Sincis I0dldz. By simple geometric principles,

one can deduce that dl is equal to sinτds where ds = adτ as shown in Fig. 2.3. Thus, the light energy incident in the area dsdz can also be expressed as

(44)

The incident energy is then subjected to continual divisions because of refractions and reflections. For a p-order scattered ray pencil of polarization X, the light energy is reduced to

SX,p= Sincε2X,p (2-10)

Suppose the scattering angle of this scattered ray pencil ranges from θ to θ + dθ as shown in Fig. 2.3. The transversal area of the scattered ray pencil, at a distance r far from the emergent point, is denoted as ds′_{dz. In fact, the transversal area of the incident ray pencil}

is so small that the width dl tends to be zero. Thus, the area of the scattered ray pencil on the cylinder can be regarded as zero, i.e. s0 → 0 (s0is illustrated in Fig. 2.3). Then, the arc

length of the scattered ray pencil ds′_{is approximated to be}

ds′ = rdθ (2-11)

Then, the light intensity of the scattered ray pencil observed at r and of polarization X can be obtained by IX,p = SX,p ds′_dz, that is IX,p= I0asin τdτdzε2_X,p rdθdz = a rI0ε 2 X,pD (2-12)

The coefficient D is called the divergence factor of a light ray and is defined for an infinite circular cylinder as

D = sin τ

dθ/dτ (2-13)

From Eqs. (2-1) and (2-2), we have dθ/dτ = ±dθ′/dτ. Considering that the divergence factor D should be positive, dθ/dτ is calculated to be

dθ dτ = dθ′ dτ = 2 |m sin τ ′_{− p sin τ|} msin τ′ (2-14)

By omitting the term I0 in Eq. (2-12) and then multiplying by kr, the dimensionless

scattered intensity in the far field is expressed by

IX,p = kaDε2X,p (2-15)

where k is the wave number of the scattered ray (k = 2π/λ). The dimensionless term ka is the size parameter which depicts the size of particle relative to the wavelength of light.

(45)

inter-ference between different scattered rays should be taken into account. Besides the amplitude discussed in the preceding part, the calculation for the interference field needs also the in-formation about the phase for each scattered ray.

For a particle illuminated by plane wave, there are three factors that affect the phase of a scattered ray: reflection, optical path and focal lines. The time factor is chosen as exp(iωt).

(1) The phase shift due to reflections ϕX,r. According to the Fresnel formulas, the

refrac-tion coefficients are always positive, which means the refracrefrac-tion does not alter the phase; while the reflection coefficients may be negative, indicating the reflection may change the sign of the amplitude and thereby introduce a phase shift of π (−1 = eiπ_).

On the other hand, when total reflection occurs, all the energy of light is reflected and the Fresnel reflection coefficients r_⊥ and r_∥ are equal to exp(iδ_⊥) and exp(iδ_∥), respectively, where the phase shifts δ_⊥and δ_∥are given by

     δ_⊥= 2 arctan ( √ sin2α−M2 t cos α ) δ_∥ =2 arctan ( √ sin2α−M2 t M2 t cos α ) (2-16)

where Mt is the refractive index of the refracted medium relative to the incident

medi-um.

A scattered ray might have undergone one or more times of reflection. Since these re-flection coefficients, being positive or negative, real or imaginary, are already included in the factor εX,pdefined in Eq. (2-8), the phase shift of a p-order scattered ray due to

reflections, ϕX,r, can be retrieved directly by the argument (phase) of εX,p:

εX,p = ∥εX,p∥ exp(iϕX,r) (2-17)

(2) The phase shift due to optical path ϕp,OP. The phase shift ϕp,OP of a scattered ray is

caused by its optical path, usually compared to a reference ray. The reference ray, free of the refraction by the particle, arrives at the center of particle in the the direction of the incident ray and then emerges in the same direction as the scattered ray.

As shown in Fig. 2.4, the externally-reflected ray (p = 0) has a shorter optical path than the reference ray, thus it has a positive phase when compared to the reference ray:

ϕ0,OP =

2π

(46)

t

'

t

a

'

t

s

O

0 p =

1 p =

2 p =

s

Fig 2.4 Schematic diagram of the virtual rays (dotted) serving as the references in calculating the optical paths of scattered rays.

where σ equals a sin τ. The scattered rays of p ≥ 1 have longer optical paths than the reference rays. For the scattered ray of p = 1, the phase shift due to optical path is −2πλms, where the negative sign indicates that the phase lags for a longer optical path.

sis the geometric length between two successive interactions, equal to 2a sin τ′_{. Then,}

its phase relative to the reference ray is calculated by ϕ1,OP = − 2π λ ms − (− 2π λ 2d) = 2π

λ(2a sin τ − 2ma sin τ

′₎

(2-19)

Then, for the scattered ray of order p, the phase shift relative to the reference ray is deduced as

ϕp,OP =

2π

λ 2a(sin τ − pm sin τ

′₎ _(2-20)

(3) The phase shift due to focal lines ϕp,FL. Because of the Gouy anomaly [144–146], at

the passage of any focal line the phase advances by π/2. Two types of focal lines are categorized by van de Hulst (the pp. 201 and 202 of [1]) for the scattering of plane wave by a sphere:

type A: Any point of intersection of two adjacent rays in a meridional cross section is a point of a focal curve.

type B: Any point where a ray intersects the axis is a point of focal line because the corresponding rays in other meridional sections have the same point of